This English version may function a textual content for a primary 12 months graduate direction on differential geometry, as did for a very long time the Chicago Notes of Chern pointed out within the Preface to the German version. compatible references for ordin­ ary differential equations are Hurewicz, W. Lectures on traditional differential equations. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, big apple, 1977. Upon David Hoffman fell the tricky activity of remodeling the tightly built German textual content into one that might mesh good with the extra secure layout of the Graduate Texts in arithmetic sequence. There are a few e1aborations and a number of other new figures were extra. I belief that the benefits of the German version have survived while whilst the efforts of David helped to explain the overall notion of the path the place we attempted to place Geometry earlier than Formalism with out giving up mathematical rigour. 1 desire to thank David for his paintings and his enthusiasm through the complete interval of our collaboration. while i want to commend the editors of Springer-Verlag for his or her endurance and sturdy suggestion. Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German variation This e-book has its origins in a one-semester direction in differential geometry which 1 have given time and again at Gottingen, Mainz, and Bonn.

Those notes include components: chosen in York 1) Geometry, New 1946, issues collage Notes Peter Lax. through Differential within the 2) Lectures on Stanford Geometry huge, 1956, Notes J. W. college by way of grey. are the following with out crucial They reproduced switch. Heinz used to be a mathematician who mathema- Hopf well-known vital tical principles and new mathematical situations.

1). A generated surface is a surface produced by a curve c(v), veI, and a one-parameter group of isometries 1': (u, v) e 1 x 1. f(u, v) = 1'(u, c(v)), It is certainly possible thatfis not a regular map, so one needs to as sume additional conditions to insure thatfis a surface. Some examples of generated surfaces are the sphere, the torus, and, more generally, any surface of revolution. An example of a generated surface which is not a surface of revolution is given by the helicoid. Let c(v) = (v, O, O), ve IR, and let 1'(t, x) = (Xl +r cos t sin t, _Xl sin t + x 2 cos t, r + bt), b i= O, te IR.

I) Letf: U -+ 1R3 be a surface. Let U E U. The inner product on 1R3 ;:;: T f (u)1R3 induces a quadratic form on Tuf c T/(U)1R3 ;:;: 1R3 by restriction. This form is called thefirstfundamentalform and is denoted sometimes by g or gu and sometimes by I or lu. ii) The inner product on Tf(U)1R3 ;:;: 1R3 composed with the linear map dfu: 1R2 ;:;: T uIR 2-+ T f (u)1R3 ;:;: 1R3 induces a quadratic form on T uIR 2 which is also called the first fundamental form. It is also denoted by g or 1, and it will sometimes be written "df· df" Remark.

They are regular by (i). Notice that by condition (ii), not only are the surfaces orthogonal, but = O on each of them. Furthermore, Fw(u, v, wo) is normal to f Wo at (u, v, wo) (and the identical relation holds for the other two surfaces) and, ditferentiating, g12 = (Fu·Fw). ·Fw)u = O. )w Therefore Fuv·Fw each of the surfaces. 2 Proposition (Dupin). The coordinate curves on a surface in a triply orthogonal system are lines of curvature. 3 An example. Second order confocal surfaces. p(p) = - c-p y2 Z2 + b-p -- + - - 1 = O.